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Cauchy's Integral Formula

If f(z) is analytic within and on a closed curve C and a is a point within C, then
f(a) = (1/2πi)c [f(z)/(z-a)] dz

Cauchy's Integral Formula for the derivative of an analytic function

If a function f(z) is analytic in a region D, then its derivative at any point z=a of D is also analytic in D and is given by
f '(a) = (1/2πi)c [f(z)/(z-a)2] dz
where C is any closed contour in D surrounding the point z=a.

Theorem

If a function f(z) is analytic in a region D, then its derivative at any point z=a of D has derivatives of all orders, all of which are again analytic function in D, their values are given by
fn(a) = (n!/2πi)c [f(z)/(z-a)n+1] dz
where C is any closed contour in D surrounding the point z=a.

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